6 \begin{lem}[Small blobs]

     6 \begin{thm}[Small blobs]

     8 \end{lem}

     8 \end{thm}

    10 Given a blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{1, \ldots, k\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{1,\ldots,k\}} \in \bc_0(M)$, and $d b_\cS = \sum_{\cS' = \cS'\sqcup\{i\}} \text{some sign} b_{\cS'}$.

    10 We begin by describing the homotopy inverse in small degrees, to illustrate the general technique.

    11 Similarly, for a configuration of $k$ blobs $\beta$ (that is, an choice of embeddings of balls in $M$, satisfying the disjointness rules for blobs, rather than a blob diagram, which is additionally labelled by appropriate fields), $\beta_\cS$ denotes the result of erasing a subset of blobs. We'll write $\beta' \prec \beta$ if $\beta' = \beta_\cS$ for some $\cS$. Finally, for finite sequences, we'll write $i \prec i'$ if $i$ is subsequence of $i'$, and $i \prec_1 i$ if the lengths differ by exactly 1.

    11 We will construct a chain map $s:  \bc_*(M) \to \bc^{\cU}_*(M)$ and a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $\bdy h+h \bdy=\id - i\circ s$. The composition $s \circ i$ will just be the identity.

    12 

    13 On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_i x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.

    14 

    15 On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term `makes $\beta$ small', while the other term `gets the boundary right'. First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(0,1)$ is the identity and $\phi_\beta(1,0)$ makes the ball $\beta$ small. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(s,t,0)$ makes the ball $\beta$ small for all $s+t=1$, while $\phi_{\eset \prec \beta}(0,t,u) = \phi_\eset(t,u)$ and $\phi_{\eset \prec \beta}(s,0,u) = \phi_\beta(s,u)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by

    16 $$s(b) = \phi_\beta(1,0)(b) + \restrict{\phi_{\eset \prec \beta}}{u=0}(\bdy b).$$

    17 Here, $\phi_\beta(1,0)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{u=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. We now check that $s$, as defined so far, is a chain map, calculating

    18 \begin{align*}

    19 \bdy (s(b)) & = \phi_\beta(1,0)(\bdy b) + (\bdy \restrict{\phi_{\eset \prec \beta}}{u=0})(\bdy b) \\

    20 		 & = \phi_\beta(1,0)(\bdy b) + \phi_\eset(1,0)(\bdy b) - \phi_\beta(1,0)(\bdy b) \\

    21 		 & = \phi_\eset(1,0)(\bdy b) \\

    22 		 & = s(\bdy b)

    23 \end{align*}

    24 Next, we compute the compositions $s \circ i$ and $i \circ s$. If we start with a small $1$-blob diagram $b$, first include it up to the full blob complex then apply $s$, we get exactly back to $b$, at least assuming we adopt the convention that for any ball $\beta$ which is already small, we choose the families of homeomorphisms $\phi_\beta$ and $\phi_{\eset \prec \beta}$ to always be the identity. In the other direction, $i \circ s$, we will need to construct the homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ for $*=0$ or $1$. This is defined by $h(b) = \phi_\eset(b)$ when $b$ is a $0$-blob (here $\phi_\eset$ is a one parameter family of homeomorphisms, so this is a $1$-blob), and $h(b) = \phi_\beta(b) - \phi_{\eset \prec \beta}(\bdy b)$ when $b$ is a $1$-blob (here $\beta$ is the ball in $b$, and this is the action of a one parameter family of homeomorphisms on a $1$-blob, so a $2$-blob).

    25 

    26 \begin{align*}

    27 (\bdy h+h \bdy)(b) & = \bdy (\phi_{\beta}(b) - \phi_{\eset \prec \beta}{\bdy b}) + \phi_\eset(\bdy b)  \\

    28 	& = b - \phi_\beta(1,0)(b) - \phi_\beta(\bdy b) - (\bdy \phi_{\eset \prec \beta})(\bdy b) + \phi_\eset(\bdy b) \\

    29 	& =  b - \phi_\beta(1,0)(b) - \phi_\beta(\bdy b) -  \phi_\eset(\bdy b) + \phi_\beta(\bdy b) - \restrict{\phi_{\eset \prec \beta}}{u=0}(\bdy b) + \phi_\eset(\bdy b) \\

    30 	& = b - \phi_\beta(1,0)(b) - \restrict{\phi_{\eset \prec \beta}}{u=0}(\bdy b) \\

    31 	& = (\id - i \circ s)(b)

    32 \end{align*}

    33 

    34 

    35 Given a blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{1, \ldots, k\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{1,\ldots,k\}} \in \bc_0(M)$, and $d b_\cS = \sum_{\cS' = \cS'\sqcup\{i\}} \pm  b_{\cS'}$.

    36 Similarly, for a disjoint embedding of $k$ balls $\beta$ (that is, a blob diagram but without the labels on regions), $\beta_\cS$ denotes the result of erasing a subset of blobs. We'll write $\beta' \prec \beta$ if $\beta' = \beta_\cS$ for some $\cS$. Finally, for finite sequences, we'll write $i \prec i'$ if $i$ is subsequence of $i'$, and $i \prec_1 i$ if the lengths differ by exactly 1.

    25 It's not immediately obvious that it's possible to make such choices, but it follows quickly from

    50 It's not immediately obvious that it's possible to make such choices, but it follows readily from the following Lemma.

    26 \begin{claim}

    27 If $\beta$ is a collection of disjointly embedded balls in $M$, and $\varphi: B^k \to \Diff{M}$ is a map into diffeomorphisms such that for every $x\in \bdy B^k$, $\varphi(x)(\beta)$ is subordinate to $\cU$, then we can extend $\varphi$ to $\varphi:B^{k+1} \to \Diff{M}$, with the original $B^k$ as $\bdy^{\text{north}}(B^{k+1})$, and $\varphi(x)(\beta)$ subordinate to $\cU$ for every $x \in \bdy^{\text{south}}(B^{k+1})$.

    29 In fact, for a fixed $\beta$, $\Diff{M}$ retracts onto the subset $\setc{\varphi \in \Diff{M}}{\text{$\varphi(\beta)$ is subordinate to $\cU$}}$.

    52 When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$.

    30 \end{claim}

    53 

    31 \nn{need to check that this is true.}

    54 \begin{lem}

    55 \label{lem:extend-small-homeomorphisms}

    56 Fix a collection of disjoint embedded balls $\beta$ in $M$. Suppose we have a map $f :  X \to \Homeo(M)$ on some compact $X$ such that for each $x \in \bdy X$, $f(x)$ makes $\beta$ small. Then we can extend $f$ to a map $\tilde{f} : X \times [0,1] \to \Homeo(M)$ so that $\tilde{f}(x,0) = f(x)$ and for every $x \in \bdy X \times [0,1] \cup X \times \{1\}$, $\tilde{f}(x)$ makes $\beta$ small.

    57 \end{lem}

    58 \begin{proof}

    59 Fix a metric on $M$, and pick $\epsilon > 0$ so every $\epsilon$ ball in $M$ is contained in some open set of $\cU$. First construct a family of homeomorphisms $g_s : M \to M$, $s \in [1,\infty)$ so $g_1$ is the identity, and $g_s(\beta_i) \subset \beta_i$ and $\rad g_s(\beta_i) \leq \frac{1}{s} \rad \beta_i$ for each ball $\beta_i$. 

    60 There is some $K$ which uniformly bounds the expansion factors of all the homeomorphisms $f(x)$, that is $d(f(x)(a), f(x)(b)) < K d(a,b)$ for all $x \in X, a,b \in M$. Write $S=\epsilon^{-1} K \max_i \{\rad \beta_i\}$ (note that is $S<1$, we can just take $S=1$, as already $f(x)$ makes $\beta$ small for all $x$). Now define $\tilde{f}(t, x) = f(x) \compose g_{(S-1)t+1}$.

    61 

    62 If $x \in \bdy X$, then $g_{(S-1)t+1}(\beta_i) \subset \beta_i$, and by hypothesis $f(x)$ makes $\beta_i$ small, so $\tilde{f}(t, x)$ makes $\beta$ small for all $t \in [0,1]$. Alternatively, $\rad g_S(\beta_i) \leq \frac{1}{S} \rad \beta_i \leq \frac{\epsilon}{K}$, so $\rad \tilde{f}(1,x)(\beta_i) \leq \epsilon$, and so $\tilde{f}(1,x)$ makes $\beta$ small for all $x \in X$.

    63 \end{proof}